Problem:
In a magic square, the sum of the three entries in any row, column, or diagonal is the same value. The figure shows four of the entries of a magic square. Find .
Solution:
The sum of the entries in the first row, which is , equals the sum of the entries in the first column, hence the lower-left entry is . Because the sum of the entries in the diagonal that includes the upper-right corner is , the central entry must be . Because the sum of the entries in the second row is , the last entry in that row must be . Because the sum of the entries in the third column is , the last entry in that column must be . This puts , and on a diagonal. It follows that
hence .
The problems on this page are the property of the MAA's American Mathematics Competitions